Improved Upper Bounds on the Pebbling Numbers of the Blanu\v{s}a Snarks

TL;DR

This paper improves upper bounds on the pebbling numbers of the two Blanuša snarks using advanced weight function heuristics and linear programming, and confirms lower bounds with multiple verification methods.

Contribution

It introduces a refined application of Hurlbert's Weight Function Lemma, optimizing weight functions via linear programming for tighter bounds on the pebbling numbers.

Findings

Upper bounds for B_1 and B_2 are reduced to 28 and 29 respectively.

Lower bounds of 23 for both graphs are re-verified with independent methods.

Interval for B_1 shrinks from [23, 31] to [23, 28], for B_2 from [23, 30] to [23, 29].

Abstract

The two Blanu\v{s}a snarks $B_1$ and $B_2$ are 3-regular graphs on 18 vertices. Dantas, Lordelo, Niedermaier and Nogueira (Discrete Appl. Math. 361, 2025, pp. 336-346) established the first systematic bounds $23 \le \pi(B_i) \le 34$ for $i=1,2$. Bridi, Marquezino and Figueiredo (arXiv:2505.16050, 2025) then sharpened the upper side to $\pi(B_1) \le 31$ and $\pi(B_2) \le 30$ via a Weight Function Lemma heuristic. We push the upper bounds further to $\pi(B_1) \le 28$ and $\pi(B_2) \le 29$. The route is again Hurlbert's Weight Function Lemma, but applied one automorphism orbit at a time, with optimal weight functions coming from a linear program over a corpus of roughly $30{,}000$ rooted-subtree strategies per target. For the lower bound $\pi(B_i) \ge 23$ we re-derive the witnesses of Dantas et al. and re-verify them with two independent oracles: an exhaustive forward state-space search, and a sound-and-complete MILP encoding whose acyclicity constraint is motivated by the Milans-Clark No-Cycle Lemma. The interval for $B_1$ shrinks from $[23, 31]$ to $[23, 28]$, and for $B_2$ from $[23, 30]$ to $[23, 29]$.